Finite Group Databasebeta

← #10943829

Subgroups — #10943829

Order 768.

4004 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#239C211yesmaximal, characteristic, complemented
2C2×C3×#448C211yesmaximal, characteristic, complemented
3C2×#5172C214yesmaximal, complemented
4#82723C211yesmaximal, characteristic, complemented
5C2×#5036C211yesmaximal, characteristic, complemented
6#72132C211yesmaximal, characteristic, complemented
7#70267C214yesmaximal, complemented
8#69959C211yesmaximal, characteristic, complemented
9#69955C211yesmaximal, characteristic, complemented
10#32880—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×#239—24no—
13C22×#239—216no—
14C22×#239—24no—
15C22×#239—22no—
16C22×#239C2214yes—
17C23×D12—21no—
18C23×D12C2211yescharacteristic
19C23×C4×D3—21no—
20C23×C4×D3C2211yescharacteristic
21C22×C3×SD16C2212yescomplemented
22C2×C3×#121C2211yescharacteristic
23C2×#938C2212yescomplemented
24C2×#937—21no—
25C22×#213C2212yescomplemented
26C3×#448—24no—
27C2×C3×#103C2212yescomplemented
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×#835C2214yescomplemented
30#5220—21no—
31#5212C2211yescharacteristic
32C2×#830C2211yescharacteristic
33#5172—216no—
34C2×#821C2214yescomplemented
35C2×(C2×D3)⋊C8C2214yescomplemented
36#5036—24no—
37#4987C2212yescomplemented
38C2×#809C2212yescomplemented
39#4956—22no—
40C23×#41—31no—
41C2×#449—34no—
42C2×#448D311yescharacteristic, complemented
43C2×#447—31no—
44#2847—31no—
45#1788—34no—
46#1787—31no—
47#1786—31no—
48#1779—31no—
49C24×D3—21no—
50C24×D3D411yescharacteristic
51C22×C3×Q8—28no—
52C22×C3×Q8—22no—
53C22×C3×Q8—21no—
54C22×C3×Q8D414yescomplemented
55C22×C3×Q8C2312yes—
56C22×C3×D4—21no—
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C23×C3×C4C2311yescharacteristic
60C22×#67—22no—
61C23×Dic3D411yescharacteristic
62C2×#239—264no—
63C2×#239—264no—
64C2×#239—216no—
65C2×#239—216no—
66C2×#239—216no—
67C2×#239—28no—
68C2×#239—28no—
69C2×#239—28no—
70C2×#239—28no—
71C2×#239—28no—
72C2×#239D4116yescomplemented
73C22×D12—28no—
74C22×D12—216no—
75C22×D12—24no—
76C22×D12—21no—
77C22×D12—22no—
78C22×D12—28no—
79C22×D12D414yescomplemented
80C22×D12C2312yes—
81C22×C4×D3—28no—
82C22×C4×D3—216no—
83C22×C4×D3—22no—
84C22×C4×D3—22no—
85C22×C4×D3—24no—
86C22×C4×D3—28no—
87C22×C4×D3D412yes—
88C22×C4×D3C2312yes—
89C2×C3×SD16—28no—
90C2×C3×SD16—24no—
91C22×C3×C8C2312yes—
92C3×#121—22no—
93C3×#121—22no—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×#217—21no—
97C2×#217—22no—
98#938—28no—
99#937—44no—
100C2×#213—28no—
101C2×#213—24no—
102C2×(C2×D3)⋊C4—22no—
103C2×C4⋊Dic3C2312yes—
104C22×C3⋊C8C2312yes—
105C3×#103—24no—
106C3×C22⋊C8—22no—
107#835—28no—
108C3⋊#95—21no—
109C3⋊#95—22no—
110#830—22no—
111#821—28no—
112(C2×D3)⋊C8—28no—
113#809—24no—
114C22×#41—34no—
115C22×#41—62no—
116C22×#41—64no—
117C22×#41—616no—
118C22×#41—64no—
119C23×Q8C2×D311yescharacteristic
120C23×D4—31no—
121C23×D4—61no—
122C24×C4—31no—
123C24×C4—61no—
124C22×SD16C2×D312yescomplemented
125C22×D8—32no—
126C2×#121—32no—
127C2×#121C2×D311yescharacteristic
128C2×#120—61no—
129#449—616no—
130#448—24no—
131#447—64no—
132C2×#103—34no—
133C2×#103C2×D312yescomplemented
134C2×#102—34no—
135C2×#102—32no—
136C2×C22⋊C8—31no—
137C2×C22⋊C8—34no—
138C2×C22⋊C8C2×D311yescharacteristic
139#386—62no—
140#386—61no—
141#380—31no—
142#380—32no—
143C24×C3—21no—
144C24×C3C2×D411yescharacteristic
145C23×D3—28no—
146C23×D3—22no—
147C23×D3—24no—
148C23×D3—22no—
149C23×D3—24no—
150C23×D3—22no—
151C23×D3—22no—
152C23×D3—42no—
153C23×D3—44no—
154C23×D3—42no—
155C23×D3—42no—
156C23×D3C2×D412yes—
157C2×C3×Q8—216no—
158C2×C3×Q8—28no—
159C2×C3×Q8—216no—
160C2×C3×Q8—24no—
161C2×C3×Q8—22no—
162C2×C3×Q8—24no—
163C2×C3×Q8C2×D418yes—
164C2×C3×Q8C2×D414yes—
165C2×C3×D4—21no—
166C2×C3×D4—24no—
167C2×C3×D4—21no—
168C2×C3×D4—42no—
169C2×C3×D4—42no—
170C2×C3×D4—42no—
171C22×C3×C4—28no—
172C22×C3×C4—24no—
173C22×C3×C4—22no—
174C22×C3×C4—24no—
175C22×C3×C4—21no—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2411yescharacteristic
180C2×#67—44no—
181C2×#67—44no—
182C2×#67—44no—
183C22×Dic3—22no—
184C22×Dic3—24no—
185C22×Dic3—22no—
186C22×Dic3C2×D412yes—
187#239—2128no—
188#239—232no—
189#239—232no—
190#239—232no—
191#239—232no—
192C2×D12—264no—
193C2×D12—216no—
194C2×D12—216no—
195C2×D12—28no—
196C2×D12—216no—
197C2×D12—216no—
198C2×D12—24no—
199C2×D12—24no—
200C2×D12—22no—
201C2×D12—216no—
202C2×D12C2×D414yes—
203C2×D12C2×D418yes—
204C2×C4×D3—264no—
205C2×C4×D3—216no—
206C2×C4×D3—216no—
207C2×C4×D3—216no—
208C2×C4×D3—216no—
209C2×C4×D3—24no—
210C2×C4×D3—24no—
211C2×C4×D3—22no—
212C2×C4×D3—24no—
213C2×C4×D3—22no—
214C2×C4×D3—216no—
215C2×C4×D3—24no—
216C2×C4×D3C2×D418yes—
217C3×SD16—48no—
218C2×C3×C8—24no—
219C2×C3×C8—22no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C22⋊C4—42no—
223#217—42no—
224#217—44no—
225#213—48no—
226(C2×D3)⋊C4—44no—
227C4⋊Dic3—24no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—24no—
230C25—31no—
231C25—61no—
232C2×#41—316no—
233C2×#41—68no—
234C2×#41—68no—
235C2×#41—68no—
236C2×#41—616no—
237C2×#41—616no—
238C2×#41—664no—
239C2×#41—664no—
240C2×#41—616no—
241C2×#41—68no—
242C2×#41—68no—
243C22×Q8—28no—
244C22×Q8—22no—
245C22×Q8—21no—
246C22×Q8#6714yescomplemented
247C22×Q8C22×D312yes—
248C22×D4—21no—
249C22×D4—34no—
250C22×D4—32no—
251C22×D4—68no—
252C22×D4—64no—
253C22×D4—616no—
254C22×D4—68no—
255C22×D4—62no—
256C22×D4—61no—
257C22×D4—62no—
258C22×D4C22×D311yescharacteristic
259C23×C4—21no—
260C23×C4—32no—
261C23×C4—31no—
262C23×C4—32no—
263C23×C4—68no—
264C23×C4—64no—
265C23×C4—616no—
266C23×C4—68no—
267C23×C4—62no—
268C23×C4—62no—
269C23×C4C22×D311yescharacteristic
270C2×SD16—28no—
271C2×SD16—24no—
272C2×D8—64no—
273C2×D8—68no—
274C22×C8—32no—
275C22×C8C22×D312yes—
276#121—22no—
277#121—22no—
278#121—68no—
279#120—124no—
280C2×C4⋊C4—32no—
281C2×C4⋊C4C22×D311yescharacteristic
282C2×C22⋊C4—21no—
283C2×C22⋊C4—62no—
284C2×C22⋊C4—62no—
285C2×C22⋊C4—61no—
286#103—24no—
287#103—68no—
288#102—64no—
289#102—68no—
290C22⋊C8—22no—
291C22⋊C8—68no—
292C22⋊C8—62no—
293#95—62no—
294#95—61no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—42no—
298C23×C3—41no—
299C23×C3—41no—
300C23×C3—41no—
301C23×C3—42no—
302C23×C3#12012yes—
303C23×C3#13512yes—
304C23×C3AGL(1,ℤ/8)12yes—
305C23×C3C22×D411yescharacteristic
306C22×D3—28no—
307C22×D3—216no—
308C22×D3—22no—
309C22×D3—22no—
310C22×D3—24no—
311C22×D3—28no—
312C22×D3—22no—
313C22×D3—22no—
314C22×D3—22no—
315C22×D3—22no—
316C22×D3—48no—
317C22×D3—48no—
318C22×D3—416no—
319C22×D3—44no—
320C22×D3—48no—
321C22×D3—416no—
322C22×D3—42no—
323C22×D3—42no—
324C22×D3—44no—
325C22×D3—44no—
326C22×D3—48no—
327C22×D3#12014yes—
328C3×Q8—216no—
329C3×Q8—24no—
330C3×Q8—28no—
331C3×Q8#12018yescomplemented
332C3×D4—42no—
333C3×D4—42no—
334C3×D4—84no—
335C2×C3×C4—216no—
336C2×C3×C4—24no—
337C2×C3×C4—28no—
338C2×C3×C4—22no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4#12012yes—
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C22×D414yes—
347C2×C3×C4C22×D411yescharacteristic
348#67—88no—
349C2×Dic3—28no—
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3—42no—
355C2×Dic3#12014yes—
356D12—264no—
357D12—216no—
358D12—28no—
359D12—24no—
360D12#12018yescomplemented
361C4×D3—264no—
362C4×D3—216no—
363C4×D3—28no—
364C4×D3—28no—
365C3×C8—42no—
366C3⋊C8—42no—
367C24—21no—
368C24—32no—
369C24—64no—
370C24—68no—
371C24—64no—
372C24—62no—
373C24—62no—
374C24—62no—
375C24—62no—
376C24—122no—
377C24—124no—
378C24—122no—
379C24—122no—
380C24D3×D411yescharacteristic
381#41—632no—
382#41—6128no—
383#41—632no—
384#41—632no—
385#41—632no—
386C2×Q8—216no—
387C2×Q8—28no—
388C2×Q8—216no—
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—24no—
392C2×Q8D3×D418yes—
393C2×Q8C2×#6714yes—
394C2×D4—21no—
395C2×D4—24no—
396C2×D4—21no—
397C2×D4—34no—
398C2×D4—38no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—616no—
403C2×D4—616no—
404C2×D4—664no—
405C2×D4—616no—
406C2×D4—616no—
407C2×D4—68no—
408C2×D4—616no—
409C2×D4—64no—
410C2×D4—64no—
411C2×D4—62no—
412C2×D4—124no—
413C2×D4—124no—
414C2×D4—124no—
415C22×C4—28no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—32no—
421C22×C4—38no—
422C22×C4—64no—
423C22×C4—616no—
424C22×C4—64no—
425C22×C4—664no—
426C22×C4—616no—
427C22×C4—616no—
428C22×C4—616no—
429C22×C4—616no—
430C22×C4—64no—
431C22×C4—62no—
432C22×C4—64no—
433C22×C4—62no—
434C22×C4—64no—
435C22×C4—62no—
436C22×C4—62no—
437C22×C4D3×D411yescharacteristic
438C22×C4D3×D412yes—
439C22×C4C2×#6712yes—
440C22×C4C23×D311yescharacteristic
441SD16—48no—
442D8—128no—
443C2×C8—24no—
444C2×C8—22no—
445C2×C8—62no—
446C2×C8—64no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—64no—
450C22⋊C4—42no—
451C22⋊C4—124no—
452C22⋊C4—124no—
453C22⋊C4—122no—
454C22×C3—22no—
455C22×C3—24no—
456C22×C3—22no—
457C22×C3—24no—
458C22×C3—81no—
459C22×C3—81no—
460C22×C3—82no—
461C22×C3—81no—
462C22×C3—82no—
463C22×C3#44914yescomplemented
464C22×C3C2×#12011yescharacteristic
465C22×C3C2×#12012yes—
466C22×C3C2×#13511yescharacteristic
467C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
468C22×C3#57512yes—
469C2×D3—24no—
470C2×D3—28no—
471C2×D3—24no—
472C2×D3—24no—
473C2×D3—416no—
474C2×D3—432no—
475C2×D3—42no—
476C2×D3—44no—
477C2×D3—42no—
478C2×D3—48no—
479C2×D3—42no—
480C2×D3—44no—
481C2×D3—42no—
482C2×D3—48no—
483C3×C4—28no—
484C3×C4—22no—
485C3×C4—81no—
486C3×C4C2×#12012yes—
487C3×C4C2×#12012yes—
488Dic3—24no—
489Dic3—82no—
490C23—22no—
491C23—22no—
492C23—34no—
493C23—41no—
494C23—41no—
495C23—42no—
496C23—41no—
497C23—42no—
498C23—616no—
499C23—68no—
500C23—62no—
501C23—62no—
502C23—62no—
503C23—62no—
504C23—64no—
505C23—62no—
506C23—68no—
507C23—62no—
508C23—128no—
509C23—128no—
510C23—124no—
511C23—128no—
512C23—1216no—
513C23—1216no—
514C23—124no—
515C23—128no—
516C23—122no—
517C23—122no—
518C23—124no—
519C23#91412yes—
520C23#91612yes—
521C23#93712yes—
522C23C2×D3×D411yescharacteristic
523Q8—216no—
524Q8—24no—
525Q8—28no—
526Q8#93718yescomplemented
527D4—38no—
528D4—42no—
529D4—42no—
530D4—664no—
531D4—616no—
532D4—68no—
533D4—64no—
534D4—84no—
535D4—248no—
536C2×C4—216no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—34no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—664no—
547C2×C4—616no—
548C2×C4—68no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—68no—
552C2×C4—68no—
553C2×C4—122no—
554C2×C4—122no—
555C2×C4—122no—
556C2×C4#93712yes—
557C2×C4C2×D3×D414yes—
558C2×C4C2×D3×D411yescharacteristic
559C2×C4C22×#6711yescharacteristic
560C8—42no—
561C8—122no—
562C2×C3—24no—
563C2×C3—161no—
564C2×C3#284611yescharacteristic
565C2×C3C2×#44911yescharacteristic
566C2×C3#300212yes—
567C2×C3#300511yescharacteristic
568C2×C3#300612yes—
569D3—44no—
570D3—48no—
571C22—22no—
572C22—24no—
573C22—24no—
574C22—22no—
575C22—64no—
576C22—64no—
577C22—68no—
578C22—64no—
579C22—81no—
580C22—81no—
581C22—82no—
582C22—82no—
583C22—81no—
584C22—1216no—
585C22—1232no—
586C22—122no—
587C22—124no—
588C22—122no—
589C22—124no—
590C22—122no—
591C22—128no—
592C22—122no—
593C22—128no—
594C22#517214yescomplemented
595C22D3×#12012yes—
596C22C2×#91411yescharacteristic
597C22C2×#91611yescharacteristic
598C22#577712yes—
599C22C2×#93711yescharacteristic
600C4—28no—
601C4—22no—
602C4—64no—
603C4—81no—
604C4—242no—
605C4D3×#12012yes—
606C4C2×#93712yes—
607C3#3288011yescharacteristic, complemented, Sylow
608C2—24no—
609C2—124no—
610C2—128no—
611C2—161no—
612C2#8003812yes—
613C2#8004012yes—
614C2#8261211yescharacteristic
615C2C2×#517211yescharacteristic
616C2#8398011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#448C2
socleC23×C3C22×D4
hypercenterC2×#448D3
hypocenterC3#32880
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#448

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C23×#239C2semidirect
2. C2×C3×#448C2semidirect
3. C2×#5172C2semidirect
4. #82723C2semidirect
5. C2×#5036C2semidirect
6. #72132C2semidirect
7. #70267C2semidirect
8. #69959C2semidirect
9. #69955C2semidirect
21. C22×C3×SD16C22semidirect
23. C2×#938C22semidirect
25. C22×#213C22semidirect
27. C2×C3×#103C22semidirect
29. C2×#835C22semidirect
34. C2×#821C22semidirect
35. C2×(C2×D3)⋊C8C22semidirect
37. #4987C22semidirect
38. C2×#809C22semidirect
42. C2×#448D3semidirect
54. C22×C3×Q8D4semidirect
72. C2×#239D4semidirect
79. C22×D12D4semidirect
124. C22×SD16C2×D3semidirect
133. C2×#103C2×D3semidirect
246. C22×Q8#67semidirect
331. C3×Q8#120semidirect
360. D12#120semidirect
463. C22×C3#449semidirect
526. Q8#937semidirect
594. C22#5172semidirect
607. C3#32880semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C23×#2392
C2×C3×#4482
C2×#51722
#827232
C2×#50362
#721322
#702672
#699592
#699552
#328803